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Section 2.2: Subsets of R
We consider the notions of finite and infinite sets, and the cardinality of
a set.
Reasonable goals for this section are to become familiar with these ideas
and to practice interpreting descriptions of sets that are presented in
terse mathematical notation (this means, amongst other things,
distinguishing between different kinds of brackets: {}, [ ], ( ), etc.).
Dr Rachel Quinlan
MA180/MA186/MA190 Calculus
Subsets of R
105 / 213
Finite and Infinite Sets
Definition
A set is finite if it is possible to list its distinct elements one by one, and
this list comes to an end.
A set is infinite if any attempt at listing its distinct elements continues
indefinitely.
Example
{1, 2, 3, 4, 5} is a finite set - its only elements are the integers 1,2,3,4,5,
there are five of them.
Dr Rachel Quinlan
MA180/MA186/MA190 Calculus
Subsets of R
106 / 213
Finite and Infinite Sets
Definition
A set is finite if it is possible to list its distinct elements one by one, and
this list comes to an end.
A set is infinite if any attempt at listing its distinct elements continues
indefinitely.
Example
The interval [1, 3] is an infinite set - it consists of all the real numbers
that are at least equal to 1 and at most equal to 3.
[1, 3] := {x ∈ R : 1 ≤ x ≤ 3}.
Note: The symbol “:=” here means this is a statement of the definition
of [1, 3].
Dr Rachel Quinlan
MA180/MA186/MA190 Calculus
Subsets of R
107 / 213
Finite and Infinite Sets
Definition
A set is finite if it is possible to list its distinct elements one by one, and
this list comes to an end.
A set is infinite if any attempt at listing its distinct elements continues
indefinitely.
Example
Z and Q are infinite sets.
Dr Rachel Quinlan
MA180/MA186/MA190 Calculus
Subsets of R
108 / 213
Finite and Infinite Sets
Definition
A set is finite if it is possible to list its distinct elements one by one, and
this list comes to an end.
A set is infinite if any attempt at listing its distinct elements continues
indefinitely.
Example
The set of real solutions of the equation
x 5 + 2x 4 − x 2 + x + 17 = 0
is a finite set. We don’t know how many elements it has, but it has at
most five, since each one corresponds to a factor of degree 1 of this
polynomial of degree 5.
Dr Rachel Quinlan
MA180/MA186/MA190 Calculus
Subsets of R
109 / 213
Finite and Infinite Sets
Definition
A set is finite if it is possible to list its distinct elements one by one, and
this list comes to an end.
A set is infinite if any attempt at listing its distinct elements continues
indefinitely.
Example
The set of prime numbers is infinite.
A pair of twin primes is a pair of primes that differ by 2: e.g. 3 and 5, 11
and 13, 59 and 61. It is not known whether the set of pairs of twin
primes is finite or infinite.
Dr Rachel Quinlan
MA180/MA186/MA190 Calculus
Subsets of R
110 / 213
Cardinality
Definition
The cardinality of a finite set S, denoted |S|, is the number of elements
in S.
Example
1
2
3
4
If S = {5, 7, 8} then |S| = 3.
|{4, 10, π}| = 3
|{x ∈ Z : π < x < 3π}| = 6.
Note: {x ∈ Z : π < x < 3π} = {4, 5, 6, 7, 8, 9}.
The cardinality of Q is infinite.
Dr Rachel Quinlan
MA180/MA186/MA190 Calculus
Subsets of R
111 / 213
Remarks
1
The notation “| · |” is severely overused in mathematics. If x is a real
number, |x| means the absolute value of x. If S is a set, |S| means
the cardinality of S. If A is a matrix |A| means the determinant of
A. It is supposed to be clear from the context what is meant.
2
Defining the concept of cardinality for infinite sets is trickier, since
you can’t say how many elements they have. We will be able to say
though what it means for two infinite sets to have the same (or
different) cardinalities.
Dr Rachel Quinlan
MA180/MA186/MA190 Calculus
Subsets of R
112 / 213
A silly example
Example
In a hotel, keys for all the guest rooms are kept on hooks behind the
reception desk. If a room is occupied, the key is missing from its hook
because the guests have it. If the receptionist wants to know how many
rooms are occupied, s/he doesn’t have to visit all the rooms to check s/he can just count the number of hooks whose keys are missing.
In this example, the occupied rooms are in one-to-one correspondence
with the empty hooks. This means that each occupied room corresponds
to one and only one empty hook, and each empty hook corresponds to
one and only one occupied room. So the number of empty hooks is the
same as the number of occupied rooms and we can count one by
counting the other.
Dr Rachel Quinlan
MA180/MA186/MA190 Calculus
Subsets of R
113 / 213
Bijections and bijective correspondence
Definition
Suppose that A and B are sets. Then a one-to-one correspondence or a
bijective correspondence between A and B is a pairing of each element of
A with an element of B, in such a way that every element of B is
matched to exactly one element of A.
Definition
Suppose that A and B are sets. A function f : A −→ B is called a
bijection if
Whenever a1 and a2 are different elements of A, f (a1 ) and f (a2 ) are
different elements of B.
Every element b of B is the image of some element a of A.
Dr Rachel Quinlan
MA180/MA186/MA190 Calculus
Subsets of R
114 / 213
Cardinality and bijective correspondence
If a bijective correspondence exists between two finite sets, they have the
same cardinality. Sometimes, in order to determine the cardinality of a
set, it is easiest to determine the cardinality of another set with which we
know it is in bijective correpsondence.
Example
How many integers between 1 and 1000 are perfect squares?
Solution: The list of perfect squares in our range begins as follows
1, 4, 9, 16, ...
One way of solving the problem would be to keep writing out successive
terms of this sequence until we hit one that exceeds 1000, and then
delete that one and count the terms that we have. This is actually more
work than we are asked to do, since we are not asked for the list of
squares but just the number of them.
Alternatively, we could notice that (31)2 = 961 and (32)2 = 1024.
So the numbers 12 , 22 , ... , (31)2 are all in the range 1 to 1000 and these
Dr Rachel Quinlan
of R
115 / 213
are the
only perfect squaresMA180/MA186/MA190
in that range,Calculus
the answer to ourSubsets
question
is
31.
Another example of bijective correspondence
This last example shows that it could be possible to know that there is a
bijective correspondence between two finite sets, without knowing the
cardinality of either of them.
Example
Show that the equations
x 3 + 2x + 4 = 0 and x 3 + 3x 2 + 5x + 7 = 0
have the same number of real solutions.
Dr Rachel Quinlan
MA180/MA186/MA190 Calculus
Subsets of R
116 / 213
x 3 + 2x + 4 = 0 and x 3 + 3x 2 + 5x + 7 = 0
Solution: One way of doing this is to demonstrate a bijective
correspondence their sets of real equations. We can write
x 3 + 3x 2 + 5x + 7 = (x 3 + 3x 2 + 3x + 1) + 2x + 6
= (x + 1)3 + (2x + 2) + 4
= (x + 1)3 + 2(x + 1) + 4.
This means that a real number a is a solution of the second equation if
and only if
(a + 1)3 + 2(a + 1) + 4 = 0
i.e. if and only if a + 1 is a solution of the first equation.
The correspondence a ←→ a + 1 is a bijective correspondence between
the solution sets of the two equations. So they have the same number of
real solutions.
Note: This number is at least 1 and at most 3. Why?
Dr Rachel Quinlan
MA180/MA186/MA190 Calculus
Subsets of R
117 / 213
Learning outcomes for Section 2.2
After studying this section you should be able to
Explain what is meant by the cardinality of a set;
Read and interpret descriptions of different subsets of R presented
using different standard notations. Decide what the elements of
these sets are and whether the sets are finite or infinite;
Explain what is meant by a bijective correspondence and give
examples to support your explanation.
Dr Rachel Quinlan
MA180/MA186/MA190 Calculus
Subsets of R
118 / 213